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epl draft Crowd Avoidance and Diversity in Socio-Economic Systems and Recommendation S. Gualdi1, M. Medo1 and Y.-C. Zhang1,2 3 1 Physics Department, University of Fribourg, CH-1700 Fribourg, Switzerland 1 0 2 WebSciencesCenter, SchoolofComputerScienceandEngineering, UniversityofElectronicScienceandTechnology 2 of China, Chengdu 610054, P. R. China n a J PACS 07.05.Kf–Data analysis: algorithms and implementation; data management 9 PACS 89.65.-s–Social and economic systems ] PACS 89.20.-a–Interdisciplinary applications of physics h p Abstract –Recommender systems recommend objects regardless of potential adverse effects of - theirovercrowding. Weaddressthisshortcomingbyintroducingcrowd-avoidingrecommendation c whereeachobject can beshared byonly alimited numberofusersorwhereobject utility dimin- o s isheswiththenumberofuserssharingit. Weuserealdatatoshowthatcontrarytoexpectations, . theintroductionoftheseconstraintsenhancesrecommendationaccuracyanddiversityeveninsys- s c temswhere overcrowding is not detrimental. Theobserved accuracy improvementsare explained i in terms of removing potential bias of the recommendation method. Wefinally propose a way to s y model artificial socio-economic systems with crowd avoidance and obtain first analytical results. h p [ 1 v Introduction. – Recommender systems are a pow- tween these two extremes where an object can be rec- 7 erful tool which nowadays helps most online retailers to ommended to an intermediate number of users. There is 8 8 make effective offers to their consumers. They use past alsoacloseconnectionwiththe combinatorialassignment 1 userpreferencesto recommendnew objectsthatthe users problem where agents (users) can perform certain tasks 1. might like. Research of recommendation grows rapidly (objects) with a certain cost and one searches for a bijec- 0 and tackles issues like recommendation algorithms [1,2], tive agent-task matching that minimizes the sum of the 3 recommendation in social systems [3], and the use of rec- corresponding costs [8]. We use here some of the algo- 1 ommendation in e-commerce [4]. rithms originally developed for the assignment problem. : v While there are situations where an arbitrary number Note that crowd avoidance is a general concept which Xi of users can be recommended the same object, in other can be used also in situations where resources can be situations this is not the case. For example, one cannot shared by an arbitrary number of parties and user sat- r a recommend the same restaurant to many people as it has isfaction does not decay with the number of other users limited space and service capabilities. Similarly, it is not sharing the resource. Given a set of user score (or cost) advantageoustousedataonindustrialproductionincoun- values, one can always apply an occupancy constraint or tries[5–7]andrecommendthesamenewproducttomany penalty and see how this impacts the assignment of ob- countriesasitcouldleadtounduecompetitionandapoor jects to users. It is of particular importance to note that ultimate outcome. this new assignmentis bound to be more diversethanthe We propose crowd-avoidingrecommendation which ad- originalone where no additionalconstraints werepresent. dresses this issue by imposing a strict occupancy con- For example, if an individual object scores top for many straint on individual objects or by assuming that object users,itcannotbeassignedtoallofthemiftheoccupancy utility decays with the number of users sharing it. Our constraint is sufficiently strong. Other objects then have approach is linked to physics where particles occupy the to replace it and the composition of the assigned objects energetically most favorable states but are either allowed becomes more diverse. In this way, the crowd avoidance in single occupation (fermions) or obey no restrictions concept can help address one of the long standing chal- (bosons). Although for particles there are no other op- lengesofinformationfiltering: thelackofdiversity[9]and tions, we are here interested also in situations lying be- its potential adverse impact on network topologies [10]. p-1 S. Gualdi et al. In this letter, we study crowd avoidance and its effects n ),thenaturalnextstepistofindthebestassignmentof α in various socio-economic systems. We begin by defin- objects to users. Here the two important approaches are ing possible approaches to introduce crowd avoidance in user-centered optimization where users attempt to maxi- recommendation. We illustrate the use of this concept mizetheirownu˜ andglobaloptimizationwherethesum iα on empirical DVD renting data. Although one does not of all effective utility values is maximized. User-centered expect overcrowdingto be a problem there, we show that optima can be obtained by a simple process where users includingcrowdavoidanceintherecommendationprocess arriveconsecutivelyandchoosetheirmostpreferredobject can increase the accuracy of the recommendation. To ex- (MPO) or by a more complicated process where users are plainthis unexpected observation,we use simple artificial allowed to change their choice until no one has an incen- data produced by a biased recommendation method and tive to change and a Nash equilibrium is reached. These show that crowd-avoiding recommendation effectively re- two processes have their real motivations: one might be moves this bias and thus increases recommendationaccu- tempted to leave a suddenly overcrowdedbar for another racy. Finally we propose how to model artificial systems one (corresponding to the Nash equilibrium case) but if with crowdavoidance and suggest an analytical approach one has already booked a good seat in a theater, there that can be used to study them. is no reason to care how many people did their bookings after(correspondingtotheMPOcase). Sincemultipleso- Framework. – We consider a set of U users (which lutions can be found in all three optimization approaches canberealpersonsbutalsofirmsorcountries)andasetof (due to the degeneracy of both the global optimum and O objects (which can be restaurants, hotels, or sectors of Nash equilibrium and due to the dependency of MPO on industrialproduction). Wethensupposethatappreciation the users’ order of arrival), all results presented here are of object α by user i is encoded in a single-valued utility averagedover several independent realizations. u (the higherthebetter). Onecanconsideranidealized iα Although constrained occupancy and effective utility case where the true utility values are known or a case have many features in common, they allow to view the where some other information is used as a proxy for the problem from slightly different angles. The former ap- utility. For example, recommendation scores obtained by proach has an analog in the classical assignment problem a recommendation algorithm can serve this purpose—we where one wishes to find an optimal user-objects assign- shall study this in detail in the following section. ment with respect to a global energy function [8]. Fast Ourgoalnowistomodelasystemwhereforsomereason algorithms exist to find global optima in this case and it is not convenient for too many users to share the same study their relation with user-centered optimization [11]. object and thus some “repulsion” of users is in action. Tools from spin glass theory are also of use here [12,13] Reasons for this repulsion may vary and we shall discuss (in our case, however, the true glassy phase is absent as some of them in detail later on. Two approaches are at the number of local minima grows only polynomially, not handtomodeluserrepulsion. Thefirstone(whichwewill exponentially with system size). The effective utility ap- refer to as “effective utility”) postulates an effective user proach instead allows, as we shall see later, for simpler utilitythatdecreaseswiththe numberofusersn sharing α analytical treatment. objectα. Oneofthepossibleformsfortheeffectiveutility is Evidence from empirical data. – We now demon- u˜ (n )=u /nb (1) iα α iα α strate the concept of crowd avoidance on empirical where the exponent b determines the repulsion strength.1 DVD rental data released for the NetflixPrize (see When b=0,repulsionis absentand if allusersprefer one www.netflixprize.com) fromwhichwe randomly choose singleobject,theyareallfreetoherdonit. Whenb , 2,000 users who rated at least 100 objects in the origi- →∞ repulsionisextremelystrongandutility-maximizingusers nal data and 2,000 objects rated by at least 10,000 users preferobjectswith the lowestoccupancyoverobjectssat- in the original data. The resulting set contains 592,995 isfying best their personal preferences. One can say that evaluationsintheintegerratingscalefrom1to5. Forthe these two cases represent a bosonic and fermionic limit, purposeofrecommendationwithunarydata(i.e.,without respectively. The second approach(which we will refer to ratings),ratings3ormoreareinterpretedasfavorableand as ’constrained occupancy’) postulates a rigid constraint constitute515,342user-objectlinksintheunarydataset. n m implying that each object can be shared by at Note that for the present case of users renting DVDs, the α ≤ most m users. In terms of effective utility, this is equiva- conceptof crowdavoidanceis relevantmore for its ability lent to to diversify the recommended content than for some real u n m, declineofutilitywhenmanyusersshareaDVD(although iα α u˜ (n )= ≤ (2) iα α a limited number of physical DVD copies might impact (0 nα >m. users by creating waiting times for popular objects). Givenuserpreferences(representedbyutilityvaluesuiα The usual way to evaluate a recommender system is to andbyhowthisutilitydiminisheswithobjectoccupancies move10%ofthedatatoaso-calledprobesetandthenuse 1Other forms of effective utility, such as u˜iα(nα) = uiα−bnα, theremaining90%ofthedata(aso-calledtrainingset)to arepossiblebutwedonotconsiderthem here. recoverthe missing (but known) probe part [2]. We use a p-2 Crowd Avoidance and Diversity in Socio-Economic Systems and Recommendation variantofapopularandhigh-performingrecommendation 0.30 method Singular Value Decomposition (SVD) [14,15] for data with ratings and the Probability Spreading method )m (tPherotbraSi)ni[n1g6,s1e7t]isfoursdedattaowobitthaoinutesrtaitminagtse.dIrnatbinogtshocrarseecs-, n (P Pb()00..43 ucormseeamr-toeendbdjefaoctrtioepnaacishrcso.urseUesrltuiimiwαaht(eetrlheye,raahnigrkahneokrfeotdhbeljiescbtteotαfteoirsb)rjefocrt.saIinlsl precisio0.20 precision 00..2010 251/b50 75 iα traditional recommendation, top T objects from a user’s 0.10 recommendation list are “recommended”. One then eval- 0 1 2 3 10 10 10 10 uates performance of the recommendations by comparing maximal occupancy m these top T objects with user-object pairs in the probe— themoreofthemappearamongtheobjectsrecommended Fig. 1: Recommendation precision vs maximal occupancy m: to the given user, the better. The usual corresponding black and red lines denote results for data with ratings (SVD metric is called precision and it is defined as the ratio of recommendation)anddatawithoutratings(ProbSrecommen- recommendedprobeobjectstothetotalnumberofrecom- dation), respectively. The solid and dashed lines denote re- mendedobjects TU. Note that the standardprocedure of sultsforMPOandglobaloptimization,respectively. Theinset creating the probe by random selection of links results in shows precision vs 1/b with the solid and dashed lines repre- sentingMPO and Nash equilibrium, respectively. popularobjectsbeing over-representedinthe probe. This favors the ProbS method which is popularity-biased and disadvantages the SVD method. As a result, SVD under- ing task but fortunately, effective algorithms exist for the performsProbSdespitetheformerusingmoreinformation case of constrained occupancy. We use the classical Hun- (i.e., the ratings) and thus being generally more reliable. garian algorithm [11] which, due to memory constraints, Since our focus here does not lie in a direct comparison allowsustostudythesystemfortheoccupancyconstraint of these two methods, this aspect is not essential for our m 12. Duetoexcessivecomputationalcomplexityofthe analysis. ≤ problem, we do not consider global optimization for u˜ To evaluate the diversity of recommended objects, we iα given by Eq. (1). use their effective number The resulting precision dependencies are shown in O Fig. 1. When m U or b 0, the allowed occupancy 2 2 neff :=(TU) nα (3) is enough to accom≈modate a≈ll users or repulsion is weak .αX=1 and results are thus identical with the assignment of the where nα is the number of users who get recommended object with the highest uiα to each user. When m = 1 object α. When all objects are recommend equally of- or b (the fermionic limit), one is forced to assign → ∞ ten, nα = TU/O and neff = O; when the same object is much inferior objects to some users and the recommen- recommended to all users, neff =1. dation precision suffers. However, the course of precision To study the effectof crowdavoidancewe turnu into is not monotonous: when some intermediate occupancy iα u˜ (n )andusethetwoabove-describedapproaches(con- constraint is applied, precision can be improved and this iα α strained occupancy and effective utility) to obtain a new improvement is further magnified if a sophisticated opti- set of recommendation lists. Since the number of evalu- mization scheme (Nash/global) is used. ated top objects T is found to be of little importance in Fig. 2a,b show the effective number of objects recom- our tests, we set T = 1 for simplicity (thus, only one ob- mendedtousers,neff,whichgrowswithrepulsionstrength jectisrecommendedtoeachuser). Thesimplestapproach as expected. At m = 14 which maximizes the preci- to select the recommended objects is based on local opti- sion value P(m) for SVD recommendation, the observed mization: users arrive in random order and choose their neff 155 is significantly higher than neff 11 achieved ≈ ≈ most preferred object (MPO), that is the one with the when the occupancy constraint is missing. We can thus highest present effective utility. Since the outcome de- concludethattheartificialoccupancyconstraintallowsus pends on the order of arrival of users, we always average to simultaneously improve precision and diversity of rec- ourresultsover1,000independentrealizationsofthispro- ommendationlists. Furthermore,onecaneasilyshowthat cess. It is straightforward to generalize this approach to the observedprecisionimprovementis notjustanartifact a Nash equilibrium framework where user preferences are of using recommendation methods which do not put best evaluated repeatedly and users are allowed to switch to objects to the top of their recommendation lists. Fig. 2c another object which they prefer more than their origi- shows precision P(r) when objects ranked r are recom- nally selected object. Finally, it is also possible to con- mended to each user without any regards to occupancy sider global optimization where a global quantity, in our andrepulsionanddemonstratesthatthefurtherdownthe case the total effective utility, is maximized. Finding a recommendation list we go, the lower the achieved preci- globallyoptimalassignmentofobjectstousersisadaunt- sion. Why then crowd avoidance helps? p-3 S. Gualdi et al. 3 3 0.28 10 (a) 10 (b) logarithmic Q(r) 2 2 neff11001 neff11001 )m0.26 epxopwoenre-lnatwia lQ Q(r(r)) (P 1001maxim1a0l occu1p0a0ncy m1000 1000 20 401/b60 80 100 cision e0.24 pr 0.25 1200 (c) (d) with bias 0.20 without bias 800 pr()00..1105 〈〉kr() 400 0.220 2 maxim4al occup6ancy m 8 10 0.05 0.00 0 1 10 100 1000 1 10 100 1000 Fig.3: ResultsasinFig.1fortheartificialdatadescribedinthe rank r rank r text with and without the popularity bias. The probe object occurrence probability decay as: Q(r)=p(1)[1−lnr/lnrmax] Fig.2: Panels(a)and(b)showtheeffectivenumberofrecom- (solid lines, Q(r)=0 for r ≥ rmax), Q(r)=[p(1)−ǫ1]/r+ǫ1 mended objects vs maximal occupancy and 1/b, respectively. (dashed lines), and Q(r)= [p(1)−ǫ2]e1−r +ǫ2 (dotted lines) Panels(c)and(d)showtheaveragerecommendationprecision where p(1) = 0.25. Parameters rmax, ǫ1, and ǫ2 are set in andtheaverageobjectdegreevsobjectrankintherecommen- ordertohaveanaverageof10probeobjectsforeachuser,i.e., dationlists,respectively. Redlinesrefertoresultsobtainedby P Q(r)=10. With bias, P(m) shows a maximum at m<U r ProbS and black lines refer to those obtained bySVD. in all threecases. When sufficiently strong occupancy restriction is ap- isamonotonicallydecreasingfunctionofrankr;(ii)Inthe plied, one is forced to move in recommendation lists of opposite case, a non-probe object is chosen from the re- some users from the top rank 1 to a lower rank r. This maining non-probe objects with probability proportional movedoesnotinfluencetheprecisionifthetop-rankedob- to h . In effect, the first rule states that the recommen- α ject and the newly selected object are both probe or both dationmethod used to build the lists workswelland puts non-probe. Thereforetoobserveaprecisionimprovement, probeobjectspreferentiallyatthe topofrecommendation the probability of exchanging a non-probe object at rank lists. The second rule implies that errors of the recom- 1 for a probe object at rank r must be greater than the mendation method are biased by the hidden variable h . α probability of exchanging a probe object at rank 1 for To check whether the second assumption is necessary, we a non-probe object at rank r. Such a situation can oc- also present results for the case where non-probe objects cur if the used recommendationmethod is biased in some are chosen purely at random. way and, along with successful recommendations demon- Artificial datasets were created for 2,000 users and stratedbyFig.2c,placessomewrongobjectsatthetopof 2,000 objects as in our real data. h values were drawn α many users’ recommendation lists. For example, Fig. 2d from the standard log-normal distribution with mean 0 shows that ProbS is strongly biased towards popular ob- andsigma1. WethenrunMPOforartificialdatasetsand jects that tend to end up at the top of recommendation averageovermultiplerealizationsofrecommendationlists lists(atthe sametime, SVDis notpopularitybiasedorit and the order of arrival of users. Fig. 3 shows that when is even weakly biased in the opposite direction). bias is present, constrained occupancy can enhance the The question still remains of how such advantageous recommendation precision. The shape and magnitude of channeling and recommendation bias arise. To approach this enhancement depends on Q(r) (note that in particu- it, we construct a set of artificial recommendation lists larthe logarithmically-decayingQ(r)issupportedbyhow producing the same phenomenon. Each artificial object recommendation precision decays when only objects of a is assigned a random hidden variable h which encodes a particular ranking r are chosen—this is shown as p(r) in α particularcharacteristicoftheobject(popularityorsome- Fig. 2c) and on the distribution of h (the narrower the α thing less tangible) to which a recommendation method distribution,theweakerthebiasandthesmallerthepossi- can be sensitive and biased. For example, the general ble gaindue to constrainedoccupancy). When the bias is bias of recommender systems towards popular objects is removed entirely, no precision improvements can be seen well known and represents one of open problems in this and P(m) grows monotonically with m (i.e., the weaker field [17,18]. A recommendation list for a given user con- theoccupancyconstraint,the higherthe precision). Simi- tains a small number of probe objects which are chosen larly, if the choice of both probe and non-probe objects is fromallobjectsatrandomandthe restarenon-probeob- subject to the same bias, precision improvement vanishes jects. The ranking of all objects is then constructed from too (the two groups of objects then essentially merge). If toptobottombyapplyingtwosimplerules: (i)Withprob- probeandnon-probeobjectsaresubjecttodifferentkinds ability Q(r), a probe object is chosenat randomfromthe of bias, precision improvement can be still achieved. remaining probe objects and placed at rank r; Here Q(r) To summarize, we found that if a recommendation al- p-4 Crowd Avoidance and Diversity in Socio-Economic Systems and Recommendation gorithmhas some bias,then introducinguser repulsionor 10-2 constrained occupancy not only helps to increase the diversity of recommendations but it may also enhance their -3 10 precision. This enhancement can be substantial if the ac- tualbiasdisagreeswithpreferencesofthe usersbutitcan ν) -4 alsovanishifthetwoareinline(e.g.,ifanalgorithmisbi- f(10 ased towards popular objects and the users want popular objects). Note that we studied data where no real prefer- 10-5 enceforlowobjectoccupancyisexpected. Intheopposite case, one could expect even stronger positive effects from -6 10 -2 -1 0 1 2 introducing crowd avoidance. 10 10 10 10 10 ν= n/〈n〉 One can generalize Eq. (2) to a heterogeneous occu- pancyconstraintbyimposingn m . Forexample,one α α can set m =Ckl where k is po≤pularity (degree) of ob- Fig. 4: Degree distribution for three values of the repulsion α α α parameter b: b =0.5 (red squares), b =0.25 (orange crosses). ject α in the input data, exponent l determines the level b=0.1(bluecircles). DashedblacklinescorrespondtoEq.(5) ofheterogeneity,andC allowsfor fixing the averagemax- (valuesof F are respectively 2.5, 3.0 and 4.5). Thesimulation imum occupancy at a given value m. l = 0 corresponds parameters are c=0.5, U =5·105 and O=103; hni=U/O. to the already discussed case where all m =m. Prelimi- α nary results for the SVD method show that with l > 0 it is possible to further magnify the precision improvement arrive and select the object with the currently highest ef- toalevelsimilartothatofProbS(thatis,precisionvalues fective utility. This flattens the effective utility landscape around0.25). Thisisinlinewiththepreviousobservation and the effective utility qα/nbα is thus constant across all that SVD suffers of the way the probe set is selected. If objects in the limit of a large number of users, imply- popular objects canbe sharedby more users thanunpop- ing qα = A(nα/ n )b where A is determined by requiring h i ular ones (i.e., when l > 0), they are more likely to be αnα = U (each user selects one object). When user recommended and thus the precision can be further im- preferences are not perfectly correlated, a similar process P proved. For the ProbSmethod, we observeno substantial takes place but the situation is more complicated. When precision improvement for l=0. a user selects an object, not only the intrinsic quality qα 6 but also the individual user preferences x matter. One iα Analysis of artificial data. – Inadditionto the de- canillustrate their effect on an example of an object with scribed practical applications, a complementary view can q = 0 which is obviously chosen by no user when c = 1. α be obtained by studying artificial systems where crowd When c < 1, positive x terms make effective utility of iα avoidance plays a role. To do that, we replace empirical this objectpositivefor someusers andn >0. This leads α user preferences with artificial correlated utility values us to a generalizeddependency between q and n which α α hastheformq +F =A(n / n )b whereF reflectsfluctu- uiα =√1 cxiα+√cqα (4) α α h i − ations of u around q and A again makes it possible to iα α where qα represents the intrinsic quality of object α, xiα achieve αnα =U. Sincethedistributionofqαisknown, quantifiesthe individualpreferencesofuseriforobjectα, this relation directly leads to the degree distribution P and c [0,1] determines the magnitude of user-user cor- ∈ bAνb−1 (Aνb F)2 relations. Elements xiα and qα are drawn independently f(ν)= exp − (5) from the standard normal distribution (0,1) and thus 2π(1 c) − 2(1 c) N − (cid:16) − (cid:17) the distribution of u is also (0,1) independently of iα N where ν := n/ np= nU/O. One can approximate F by the value of c. The Pearson correlation of scores given to h i extreme statistics for a normally distributed variable [19] an object by two different users is C(i,j) = c. We focus or by relating it to the fluctuation magnitude occurring here on the case represented by Eq. ((1)) where the effec- with the probability 1/U (thus for one user on average). tive utility gradually decreases with the number of users sharing an object. Note that the term n−b in u˜ affects Sincetheseapproximationsaretoocrudetoproducegood α iα fits of the data, we choose F by hand instead and then only the magnitude not the sign of the effective utility. adjust A accordingly. Fig. 4 shows good agreement be- Objects that have negative score u for a given user are iα tween this semi-analytical result and degree distributions therefore bound not to be chosen by this user regardless followingfromnumericalMPOmatchinginasystemwith of their occupancy and occupancy of the other objects. 1,000 objects, 500,000 users, and c=0.5. The first step is to compute the resulting object degree distribution f(n ) for different levels of correlation and Discussion. – Althoughrecommending the same ob- α valuesoftherepulsionstrength. Therelationbetweenob- ject to too many users can have adverse effects in many ject utility u and degree n is particularly simple when real situations, this letter is to our best knowledge the iα α usercorrelationisperfectandthusu =q . Inthecaseof first attempt to introduce and study crowd-avoidance in iα α user-centered optimization (MPO/Nash), users gradually recommender systems. We showed that applying object p-5 S. Gualdi et al. occupancyconstraintsoruserrepulsiontoresultsofanor- portional to exp[βu ].) Various forms of user attraction iα dinary recommendation method can be beneficial in two can then manifest themselves in positive feedback mecha- complementary ways. Firstly, it naturally increases the nismssuchaspreferentialattachmentanditsvariants[21, diversity of the recommended content and thus helps to Ch. 14]. This effectively extends the fermion-boson range address one of the long standing issues in information fil- discussed here into a more complete one: fermion-boson- tering [17]. Recommendation diversity is further closely preferential attachment. connected to sustainability of information filtering tools which emerges as an important challenge [10]. Secondly, ∗∗∗ crowd avoidance can improve accuracy of the resulting Thisworkwassupportedbythe EUFET-Openproject recommendations—despite the fact that herding of users QLectives (grant no. 231200) and by the Swiss National does not reduces their benefits in the studied DVD rental Science Foundation (grant no. 200020-132253). We ac- data. 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